What is the intuition behind how word embeddings bring information to a neural network?
Shakespeare once said "A rose by any other name would smell as sweet" (Romeo and Juliet). Words are just labels we attach to ideas for convenience. By using one hot we remain tied to the letter sequence r,o,s,e, and some other structure must take on the responsibility of attaching the context of sweetness to it.

Word embeddings learn a multi-dimensional context. What exactly the context is of each dimension of the embedding is something of a mystery and simply emerges from the learning. The larger the number of dimensions, the greater the possibility that some combination of the dimensions will represent the sweetness context, but it might be quite hard to tease out.

So you can attach the idea of sweetness to one member of a one-hot structure, but it must necessarily be a part of a rules-based approach. Embeddings, when they are working well, will not need the rules.